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Square array A(n,k) read by antidiagonals: A(n,k) = T(gcd(n,k), lcm(n,k)), where T(n,k) is sequence A000027 considered as a two-dimensional table.
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%I #14 Feb 16 2025 08:33:44

%S 1,2,2,4,5,4,7,16,16,7,11,12,13,12,11,16,46,67,67,46,16,22,23,106,25,

%T 106,23,22,29,92,31,191,191,31,92,29,37,38,211,80,41,80,211,38,37,46,

%U 154,277,379,436,436,379,277,154,46,56,57,58,59,596,61,596,59,58,57,56,67,232,436,631,781,862,862,781,631,436,232,67,79,80,529,212,991,302,85,302,991,212,529,80,79

%N Square array A(n,k) read by antidiagonals: A(n,k) = T(gcd(n,k), lcm(n,k)), where T(n,k) is sequence A000027 considered as a two-dimensional table.

%C The array is read by descending antidiagonals as A(1,1), A(1,2), A(2,1), A(1,3), A(2,2), A(3,1), etc.

%H Antti Karttunen, <a href="/A286101/b286101.txt">Table of n, a(n) for n = 1..10585; the first 145 antidiagonals of array</a>

%H MathWorld, <a href="https://mathworld.wolfram.com/PairingFunction.html">Pairing Function</a>

%F A(n,k) = T(gcd(n,k), lcm(n,k)), where T(n,k) is sequence A000027 considered as a two-dimensional table, that is, as a pairing function from N x N to N.

%F A(n,k) = A(k,n), or equivalently, a(A038722(n)) = a(n). [Array is symmetric.]

%e The top left 12 X 12 corner of the array:

%e 1, 2, 4, 7, 11, 16, 22, 29, 37, 46, 56, 67

%e 2, 5, 16, 12, 46, 23, 92, 38, 154, 57, 232, 80

%e 4, 16, 13, 67, 106, 31, 211, 277, 58, 436, 529, 94

%e 7, 12, 67, 25, 191, 80, 379, 59, 631, 212, 947, 109

%e 11, 46, 106, 191, 41, 436, 596, 781, 991, 96, 1486, 1771

%e 16, 23, 31, 80, 436, 61, 862, 302, 193, 467, 2146, 142

%e 22, 92, 211, 379, 596, 862, 85, 1541, 1954, 2416, 2927, 3487

%e 29, 38, 277, 59, 781, 302, 1541, 113, 2557, 822, 3829, 355

%e 37, 154, 58, 631, 991, 193, 1954, 2557, 145, 4006, 4852, 706

%e 46, 57, 436, 212, 96, 467, 2416, 822, 4006, 181, 5996, 1832

%e 56, 232, 529, 947, 1486, 2146, 2927, 3829, 4852, 5996, 221, 8647

%e 67, 80, 94, 109, 1771, 142, 3487, 355, 706, 1832, 8647, 265

%o (Scheme)

%o (define (A286101 n) (A286101bi (A002260 n) (A004736 n)))

%o (define (A286101bi row col) (A000027bi (gcd row col) (lcm row col)))

%o (define (A000027bi row col) (* (/ 1 2) (+ (expt (+ row col) 2) (- row) (- (* 3 col)) 2)))

%Y Cf. A000124 (row 1 and column 1), A001844 (main diagonal).

%Y Cf. A000027, A003989, A003990, A003991, A038722, A285722, A285732, A286098, A286099, A286102, A285724.

%K nonn,tabl,changed

%O 1,2

%A _Antti Karttunen_, May 03 2017